Reciprocal graphs

Abstract

Eigenvalue of a graph is the eigenvalue of its adjacency matrix. A graph $G$ is reciprocal if the reciprocal of each of its eigenvalue is also an eigenvalue of $G$. The Wiener index $W(G)$ of a graph $G$ is defined by $W(G)=\frac{1}{2} \sum_{d \in D} d$ where $D$ is the distance matrix of $G$. In this paper some new classes of reciprocal graphs and an upperbound for their energy are discussed. Pairs of equienergetic reciprocal graphs on every $n \equiv$ $0 \bmod (12)$ and $n \equiv 0 \bmod (16)$ are constructed. The Wiener indices of some classes of reciprocal graphs are also obtained.